LR Rabiner and RW Schafer, June 3

DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009
8.3. HOMOMORPHIC ANALYSIS OF THE SPEECH MODEL 447
log e | G(e j2π FT ) |
log e | R(e j2π FT ) |
3
2
1
0
−1
−2
−3
(a) Glottal Pulse Spectrum
−4
0 1000 2000 3000 4000 5000
frequency in Hz
1
0
−1
−2
−3
(c) Radiation Load Frequency Response
−4
0 1000 2000 3000 4000 5000
frequency in Hz
log e | V(e j2π FT ) |
| P(e j2π FT ) |
2
1.5
1
0.5
0
−0.5
−1
(b) Vocal Tract Frequency Response
−1.5
0 1000 2000 3000 4000 5000
frequency in Hz
100
80
60
40
20
(d) Voiced Excitation Spectrum
0
0 1000 2000 3000 4000 5000
frequency in Hz
Figure 8.15: Log magnitude (base e of DTFTs: (a) Glottal pulse DTFT
log |G(e jω )|. (b) Vocal tract frequency response, log |V (e jω )|. (c) Radiation
load frequency response log |R(e jω )|. (d) Magnitude of DTFT of periodic excitation
|P (e jω )|.
in Figure 8.15 in corresponding locations. Note that the discrete-time Fourier
transforms are plotted as log e | · | rather than in dB (i.e., 20 log 10 | · |) as is
common elsewhere throughout this text. To convert the plots in Figure 8.15(a),
(b) and (c) to dB, simply multiply by 20 log 10 e = 8.6859. We see that the
spectral contribution due to the glottal pulse is a lowpass component that has a
dynamic range of about 6 between F = 0 and F = 5000 Hz. This is equivalent
to about 50 dB spectral falloff. Figure 8.15(b) shows the spectral contribution
of the vocal tract system. The peaks of the spectrum are approximately at
the locations given in Table 8.2 with bandwidths that increase with increasing
frequency. As depicted in Figure 8.15(c), the effect of radiation is to give a high
frequency boost that partially compensates for the falloff due to the glottal
pulse. Finally, Figure 8.15(d) shows |P (e j2πF T )| (not the log) as a function of
F . Note the periodic structure due to the periodicity of p[n]. The fundamental
frequency for Np = 80 is F0 = 10000/80 = 125 Hz. 6
Now if the components of the speech model are combined by convolution,
as defined in the upper branch of Figure 8.12, the result is the synthetic speech
signal s[n] which is plotted in Figure 8.16(a). The frequency-domain repre-
6 In order to be able to make the plot in Figure 8.15(d) in Matlab it was necessary to use
β = 0.999; i.e., the excitation was not perfectly periodic.